Physics 115 Christmas Exam - 1993

1.

A glass plate 3.00 cm thick, with an index of refraction of 1.5 and plane parallel faces, is held with its faces horizontal and its lower face 9.00 cm above a printed page. Find the position of the image of the page formed by rays making a small angle with the normal to the plate.

2.

Two springs, each with unstretched length 0.20 m but having different force constants k₁ and k₂, are attached to opposite ends of a block with mass m on a level, frictionless surface. The outer ends of the springs are now attached to two pins P₁ and P₂, 0.10 m from the original positions of the ends of the springs. Let k₁=1.00 N/m, k₂=3.00 N/m, and m=0.10 kg.

(a)

Find the length of each spring when the block is in its new equilibrium position after the springs have been attached to the pins. (3 marks)

(b)

Find the period of vibration of the block if it is slightly displaced from its new equilibrium position and released. (7 marks)

3.

A wire of mass density $\mu=0.18$ kg/m is stretched between a fixed support and a pulley, as shown in the figure. A mass of 14.0 kg is attached at the free end of the wire. The segment of wire between the fixed end and the pulley can vibrate.

(a)

The wire vibrates with a frequency of 16.0 Hz in its fundamental mode. What is the length of the wire stretched between the fixed point and the pulley? (5 marks)

(b)

What are the next two frequencies at which standing waves can be set up in the wire? Sketch the standing wave pattern corresponding to these frequencies. (2 marks)

(c)

What will be the velocity of a travelling wave on the wire? (1 mark) If the attached mass is doubled, what will be the new frequency in the fundamental mode? (2 marks)

4.

A sinusoidal wave, travelling along a string, causes the displacement of the string given by the equation

$$y(x,t) \; = \; (0.1\hbox{\rm ~m}) \; \sin \left({\pi \over 3.0} x + {\pi \over 6.0} t + {\pi \over 4.0} \right)$$

(a)

What are the values of wavelength $\lambda$, the frequency f and the amplitude for this wave? (2 marks)

(b)

What is the direction of propagation of the moving wave? (2 marks)

(c)

What is the velocity of the wave along the string? (2 marks)

(d)

What is the displacement and velocity of a point on the string at position x=1.0 m at time t=2.0 s? (2 marks)

(e)

Write down the equation for a wave which when superposed with the above wave will give zero displacement everywhere along the string at all times. (2 marks)

5.

A bat, moving at a speed of 5.0 m/s, is chasing a flying insect. The bat emits a 40.0 kHz sound. When this sound is reflected back from the insect, the bat hears a sound of frequency 40.4 kHz. At what velocity is the insect moving towards or away from the bat? The speed of sound in air is 343 m/s.

6.

(a)

A sound technician sits 7.5 m from a loudspeaker L₁ which emits 75 Watts of power in all directions. What is the maximum sound level heard by the technician? (4 marks)

(b)

Now a second loudspeaker L₂ is placed 11 m away as shown in the figure. In a hurry, the technician has connected the speakers wrongly and the speakers ended up being exactly out of phase. What are the two lowest frequencies for which he hears a maximum? The speed of sound is 343 m/s and the audible range is 20-20,000 Hz. (6 marks)

7.

Microwaves of frequency 60 GHz are normally incident on a diffraction grating comprising 10 slits cut in a metal sheet. The slits have centre-to-centre separation of 3.0 cm and an individual width of 0.7 cm.

(a)

Calculate the angles of ALL the diffraction maxima assuming that the receiver is a relatively long distance from the grating. (2 marks)

(b)

Calculate the location of the first diffraction minimum if all the slits are covered up except one. (2 marks)

(c)

Sketch the intensity as a function of angle when only 2 slits are uncovered. (2 marks)

(d)

Sketch the intensity as a function of angle when 10 slits are uncovered. (2 marks)

(e)

Explain the similarities and differences of (c) and (d). (2 marks)

8.

(a)

What are the two fixed points on the thermodynamic scale (constant volume ideal gas thermometer scale)? (2 marks)

(b)

How much heat is required to warm 6.0 kg of ice (c=2220 J/kg-K) from $-10^\circ$C to $0^\circ$C (staying as ice)? (2 marks)

(c)

How much heat is given off if you freeze 0.3 kg of water initially at $30^\circ$C (c=4190 J/kg-K and L=333,000 J/kg)? (2 marks)

(d)

If 0.3 kg of water is dropped onto a 6 kg block of ice at $-10^\circ$C, what is the final temperature when equilibrium is reached, and how much ice and/or water is there? (3 marks)

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